The present invention relates to induction logging, and more in particular to a method of providing a computed induction log. For the sake of completeness it is observed that the computed induction log is sometimes called the approximate computed response.
Induction logging is a technique to determine the true resistivity of an underground formation, it is a species of resistivity logging. With the aid of the true resistivity the water and oil saturation of the formation can be assessed and that information is valuable to the exploitation of the underground formation.
In induction logging an induction logging tool is used that comprises a transmitter coil and a receiver coil arranged axially spaced apart on a mandrel. During normal operation the induction logging tool is positioned at a logging point in a wellbore in the underground formation, and the transmitter coil is energized by an alternating current. The alternating current produces an oscillating magnetic field and that results in currents induced in the formation. These currents, in turn, generate a secondary magnetic field that contributes to the voltage induced in the receiver coil.
In one embodiment of induction logging, the component of the induced voltage that is in phase with the transmitter current is selected to obtain a signal that is approximately inverse proportional to the formation resistivity. This signal is called the apparent or measured induction log. This technique is applicable to a wireline induction logging tool, wherein the alternating current has a relatively low frequency (in the kHz range) and wherein the coils are arranged on a non-conductive mandrel.
In addition to wireline logging, it is possible to measure the formation resistivity during drilling. In this case, the mandrel is conductive and the alternating current has a high frequency (in the MHz range). In this embodiment, the induction logging tool measures the strength of the secondary magnetic field at two receiver positions. From the data measured at the two positions, the amplitude ratio and the phase shift are calculated. From the amplitude ratio and the phase shift the apparent formation resistivity is calculated.
In the specification and in the claims the expression xe2x80x9cinduction logxe2x80x9d will not only be used to refer to the signal obtained for one logging point, but also to refer to the signals obtained for several logging points and to a continuously sampled record of the signals obtained when the induction logging tool is passed through an interval of the wellbore.
The apparent or measured induction log is in general not a log that is representative of the true formation resistivity alone. This is because there are environmental effects that affect the induction log. The formation is layered so that there are so-called shoulder bed effects, wherein the log at a logging point is influenced by the presence of formation layers above and below the formation layer opposite the logging point. In addition, the induction logging tool is positioned in a wellbore filled with a wellbore fluid, for example a drilling mud, which contributes to the apparent or measured resistivity. Moreover, wellbore fluid will invade the formation layer, and this wellbore fluid or mud filtrate forms an invaded zone near the wellbore. The resistivity of the invaded zone differs from the true formation resistivity, and this is a further reason why the apparent resistivity differs from the true formation resistivity. In conclusion, the true formation resistivity cannot be obtained directly from the apparent or measured induction log.
In order to compensate for the environmental effects, inverse well logging is applied, wherein the true formation resistivity that is not accessible by direct measurement is judged from indirect evidence. Inverse well logging requires iteratively forward modeling. To do that, a modelled resistivity profile, which is a model of the formation and the wellbore positioned therein, is made. With that model and with the known properties of the induction logging tool the Maxwell equations for the electromagnetic field are solved. This gives a computed induction log, which is then compared to the measured induction log. When the computed induction log does not match the measured one, the model is adjusted, and a new induction log is computed. Adjusting the model continues until the match is obtained. The true formation resistivity is the resistivity of each formation layer beyond the invaded zone(s) as obtained with the model that matches the measured induction log.
Inverse well logging thus involves forward modelling, wherein an induction log is computed.
There is a further reason why a computed induction log is required, and that is in checking whether a logging tool configuration can be applied in a particular formation.
In order to compute the induction log, one has to solve the Maxwell equations for the electromagnetic field. These equations are given below:
xcex94X{right arrow over (E)}+∂{right arrow over (B)}/∂t=0,
xcex94.{right arrow over (D)}=q, xcex94X{right arrow over (H)}xe2x88x92∂{right arrow over (D)}/∂t={right arrow over (J)}, xcex94.{right arrow over (B)}=0.
In the above equations, {right arrow over (E)} is the electric field strength (volt/meter), {right arrow over (B)} is the magnetic flux density or magnetic induction (tesla, or weber/square meter), t is time (second), {right arrow over (D)} is electric flux density (coulomb/square meter), {right arrow over (H)} is the magnetic field strength (ampere/meter), q is volume charge density (coulomb/cubic meter) and {right arrow over (J)} is the current density (ampere/square meter).
The vector products are defined in Cartesian coordinates as follows:                               ∇          X                ⁢                  v          →                    =                                    i            1                    ⁡                      (                                                            ∂                                      v                    3                                                                    ∂                                      x                    2                                                              -                                                ∂                                      v                    2                                                                    ∂                                      x                    3                                                                        )                          +                                            i              →                        2                    ⁡                      (                                                            ∂                                      v                    1                                                                    ∂                                      x                    3                                                              -                                                ∂                                      v                    3                                                                    ∂                                      x                    1                                                                        )                          +                                                            i                →                            3                        ⁡                          (                                                                    ∂                                          v                      2                                                                            ∂                                          x                      1                                                                      -                                                      ∂                                          v                      1                                                                            ∂                                          x                      2                                                                                  )                                ⁢                      xe2x80x83                    ⁢          and                      ⁢          xe2x80x83                  ∇              .                  v          →                      =                            ∂                      v            1                                    ∂                      x            1                              +                        ∂                      v            2                                    ∂                      x            2                              +                                    ∂                          v              3                                            ∂                          x              3                                      .            
For a medium having linear isotropic electromagnetic properties and where there are no sources the following constitutive relations are assumed:
{right arrow over (B)}=xcexc{right arrow over (H)}, {right arrow over (D)}=xcex5{right arrow over (E)} and {right arrow over (J)}={right arrow over (E)}/xcfx81,
wherein xcexc is the magnetic permeability (henry/meter), xcex5 is the dielectric constant (farad/meter) and xcfx81 is the resistivity (ohm.meter).
These equations have to be solved for an induction logging tool with known properties positioned in a wellbore that extends through a layered formation that is invaded by wellbore fluid. Furthermore, the wellbore can be a deviated one.
For such a problem, there is no exact solution of the Maxwell equations in three dimensions. Numerical solutions require the use of finite difference methods or finite element methods, both methods require a large amount of computing power and time.
However, exact solutions of the Maxwell equations are known for one-dimensional models. In one dimension, there are two models: (1) a one-dimensional concentric cylinder model, which takes into account the wellbore, the invasion and the true formation resistivity, but which does not take into account the effects of the formation layering and of the orientation of the logging tool with respect to the formation layering e.g. due to a deviated wellbore, and (2) a one-dimensional layered model, which takes into account the effects of the formation layering and of the orientation of the logging tool with respect to the formation layering.
An example of the first model is described in the article xe2x80x9cGeneral formulation of the induction logging problem for concentric layers about the boreholexe2x80x9d, J R Wait, IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-22, No. 1, January 1984, pages 34-42. Examples of the solutions for the second model are described in the articles xe2x80x9cTheory of induction sonde in dipping bedsxe2x80x9d, R H Hardman, L C Shen, Geophysics, Vol. 51, No. 3, March 1986, pages 800-809 and xe2x80x9cThe response of an induction dipmeter and standard induction tools to dipping bedsxe2x80x9d, S Gianzero, S-M Su, Geophysics, Vol. 55, No. 9, September 1990, pages 1128-1140.
In the one-dimensional concentric cylinder model, it is assumed that the wellbore is positioned in an infinitely thick layer comprising two or more concentric zones, which concentric zones are concentric with the wellbore. The interface between two concentric zones corresponds to the depth of invasion of the wellbore fluid. The interfaces between the concentric zones are generally assumed to be parallel to the wellbore wall. In this model the effects of the formation layering and of the orientation of the logging tool with respect to the formation layering are not taken into account.
In the one-dimensional layered model the formation layers are considered to be homogeneous layers and the position and orientation of the induction logging tool with respect to the formation layering is correctly taken into account. In one approach, the deviated wellbore is replaced by a line, which is subjected to a transformation to arrive at an imaginary wellbore that is perpendicular to the formation layering. The solution is then transformed such that it is the solution for the deviated wellbore. The one-dimensional layered model does not take into account the effect of invasion of wellbore fluid in the formation.
It is an object of the present invention to combine the two one-dimensional models so as to be able to calculate an approximate solution of the Maxwell equations that requires less computing power and time than a solution of a three-dimensional model.
To this end the method of providing a computed induction log for an induction logging tool run through an interval of a wellbore that extends through a layered formation which formation is invaded by wellbore fluid, according to the invention comprises the steps of:
(1) computing a first response of the induction logging tool by solving for each formation layer the Maxwell equations for the electromagnetic field with known logging tool properties, wellbore diameter, wellbore fluid resistivity, diameter of the invaded zone, and resistivity of the invaded zone and of the formation layer outside the invaded zone, using a one-dimensional concentric cylinder model not taking into account the effects of formation layering and of the orientation of the logging tool with respect to the formation layering, and determining for each formation layer an equivalent resistivity that gives for a homogeneous environment a response of the induction logging tool that equals the first response; and
(2) computing for a number of logging points solutions of the Maxwell equations for the electromagnetic field with the known logging tool properties and the equivalent resistivities for the formation layers, using a one-dimensional layered model wherein the effects of the formation layering and of the orientation of the logging tool with respect to the formation layering are taken into account, to provide the computed induction log of the induction logging tool.
In an alternative embodiment, the method of providing a computed resistivity for an induction logging tool at a logging point in an interval of a wellbore that extends through a layered formation which formation is invaded by wellbore fluid, according to the invention comprises the steps of:
(1) defining in each formation layer a plurality of concentric zones, wherein the first concentric zone is the wellbore, wherein the diameters of the next concentric zones correspond to the diameters of the invaded zones of the formation layers within the interval of the wellbore, and wherein the last concentric zone is outside the invaded zone with the largest diameter;
(2) computing for the logging point, for each concentric zone a first response of the induction logging tool by solving the Maxwell equations for the electromagnetic field with the known logging tool properties, using a one-dimensional layered model wherein the effects of the formation layering and of the orientation of the logging tool with respect to the formation layering are taken into account, and wherein the resistivity of each formation layer equals the resistivity of the concentric zone of that formation layer, and determining for each concentric zone an equivalent resistivity which gives for a homogeneous formation a response of the induction logging tool that equals the first response; and
(3) computing solutions of the Maxwell equations for the electromagnetic field with known logging tool properties and the diameters and the equivalent resistivities of the concentric zones, using a one-dimensional concentric cylinder model not taking into account the effects of formation layering and of the orientation of the logging tool with respect to the formation layering to provide the computed resistivity for the induction logging tool at the logging point.